Brauer-Thrall Type Theorems for Derived Module Categories

The numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of a complex (an algebra) are introduced. Cohomological range leads to the concepts of derived bounded algebra and strongly derived unbounded algebra naturally. The first and second Brauer-Thrall type theorems for the bounded derived category of a finite-dimensional algebra over an algebraically closed field are obtained. The first Brauer-Thrall type theorem says that derived bounded algebras are just derived finite algebras. The second Brauer-Thrall type theorem says that an algebra is either derived discrete or strongly derived unbounded, but not both. Moreover, piecewise hereditary algebras and derived discrete algebras are characterized as the algebras of finite global cohomological width and the algebras of finite global cohomological length respectively.